Abstract
In this paper, we construct an iterative method with memory based on the Newton–Secants method to solve nonlinear equations. This proposed method has fourth order convergence and costs only three functions evaluation per iteration and without any evaluation of the derivative function. Acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newtons interpolation polynomial of third degree. The order of convergence is increased from 4 to 5.23 without any extra function evaluation. This method has the efficiency index equal to \(5.23^{\frac{1}{3}}\approx 1.7358\). We describe the analysis of the proposed method along with numerical experiments including comparison with the existing methods. Finally, the attraction basins of the proposed method are shown and compared with other existing methods.
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References
Amat, S., Busquier, S.: On a Steffensen’s type method and its behavior for semismooth equations. Appl. Math. Comput. 177, 819–823 (2006)
Amat, S., Busquier, S.: A two-step Steffensen’s method under modified convergence conditions. J. Math. Anal. Appl. 324, 1084–1092 (2006)
Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. J. Scientia 10, 3–35 (2004)
Amat, S., Busquirer, S., Plaza, S.: Dynamics of a family of third-order itrative methods that do not require using second derivatives. J. Appl. Math. Comput. 154, 735–746 (2004)
Cordero, A., Torregrosa, J.R.: Variants of Newton method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)
Ezquerro, J.A., Grau-Sánchez, M., Hernández-Verón, M.A., Noguera, M.: A study of optimization for Steffensen-type methods with frozen divided differences. SeMA J. 70, 23–46 (2015)
Ferrara, M., Sharifi, S., Salimi, M.: Computing multiple zeros by using a parameter in Newton-Secant method. SeMA J. (2016). doi:10.1007/s40324-016-0074-0
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 634–651 (1974)
Liu, Z., Zheng, Q., Zhao, P.: A variant of Steffensens method of fourth-order convergence and its applications. Appl. Math. Comput. 216, 1978–1983 (2010)
Lotfi, T., Sharifi, S., Salimi, M., Siegmund, S.: A new class of three-point methods with optimal convergence order eight and its dynamics. Numer. Algor. 68, 261–288 (2015)
Petković, M.S., Neta, B., Petković, L.D., Džunić, J.: Multipoint methods for solving nonlinear equations. Elsevier/Academic Press, Amsterdam (2013)
Salimi, M., Lotfi, T., Sharifi, S., Siegmund, S., Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics, Int. J. Comput. Math. (2016) published online. doi:10.1080/00207160.2016.1227800
Sharifi, S., Ferrara, M., Salimi, M., Siegmund, S.: New modification of Maheshwari method with optimal eighth order of convergence for solving nonlinear equations. Open Math. (formerly Central European Journal of Mathematics) 14, 443–451 (2016)
Sharifi, S., Siegmund, S., Salimi, M.: Solving nonlinear equations by a derivative-free form of the King’s family with memory. Calcolo 53, 201–215 (2016)
Sharifi, S., Salimi, M., Siegmund, S., Lotfi, T.: A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations. Math. Comput. Simulat. 119, 69–90 (2016)
Sharma, J.R., Guha, R.K., Gupa, P.: Some efficient derivative free methods with memory for solving nonlinear equations. Appl. Math. Comput. 219, 699–707 (2012)
Traub, J.F.: Iterative methods for the solution of equations. Prentice Hall, Englewood Cliffs (1964)
Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intelligencer 24, 37–46 (2002)
Vrscay, E.R., Gilbert, W.J.: Extraneous fixed points, basin boundaries and chaotic dynamics for Schroder and Konig rational iteration functions. Numer. Math. 52, 1–16 (1988)
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)
Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)
Zheng, Q., Zhao, P., Huang, F.: A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs. Appl. Math. Comput. 217, 8196–8203 (2011)
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The present research was supported by Putra Grant Vot, No. 9442300, Universiti Putra Malaysia.
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Nik Long, N.M.A., Salimi, M., Sharifi, S. et al. Developing a new family of Newton–Secant method with memory based on a weight function. SeMA 74, 503–512 (2017). https://doi.org/10.1007/s40324-016-0097-6
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DOI: https://doi.org/10.1007/s40324-016-0097-6